optimal sizing of a stand-alone photovoltaic systems

Revue des Energies Renouvelables Vol. 18 N°2 (2015) 179 – 191

179

Optimal sizing of a stand-alone photovoltaic systems

under various weather conditions in Algeria

S. Kebaïli 1 and H. Benalla 2

1 Department of Electrical Engineering, Oum El Bouaghi University, Algeria 2 Department of Electrotechnics, Contantine 1 University, Algeria

(reçu le 02 Novembre 2014 – accepté le 30 Mai 2015)

Abstract - This paper presents an analytical method for technical-economic optimization

of a stand-alone photovoltaic system. The loss of load probability concepts for the

reliability and the total system cost for the economic criteria. The main objective of this

study is to find the optimum PV generator area and useful battery storage capacity of a

stand-alone photovoltaic system. Mathematical equations have been formulated and

LOLP curves have been constructed. A set of configuration meeting the desired LOLP are

obtained. The configuration with the minimum cost gives the optimal one. A case study

has been presented to determine the optimal sizing of a stand-alone photovoltaic system

for eight sites located at Algeria and to analyze the impact of different parameters on the

system size. This proposed analytical method is rational in terms of reliability and cost

and simple to implement for the size optimization of a stand-alone photovoltaic system at

any geographical location.

Résumé – Cet article présente une méthode analytique pour l’optimisation technico –

économique d’un système photovoltaïque autonome. Le concept de délestage de la

consommation (LOLP) pour la fiabilité et le coût total du système comme un critère

économique. L’objectif de cette étude est de trouver la surface d’un générateur PV et la

capacité utile de la batterie optimale pour un système photovoltaïque autonome. Des

équations mathématiques ont été formulées et des courbes de LOLP ont été construites.

Un ensemble de configuration pour le LOLP désirée a été obtenu. La configuration

optimale est celle obtenue avec un coût minimale. Le cas étudié pour déterminer la taille

optimale d’un système photovoltaïque autonome pour huit sites cités en Algérie et

d’analyser l’influence de différents paramètres sur la taille du système. La méthode

analytique proposée est efficace en termes de fiabilité et de coût, et simple d’être

implémenter pour l’optimisation de la taille d’un système photovoltaïque autonome à

n’importe quel site.

Keywords: Photovoltaic system – Modelling – Simulation - Optimal size – Optimizations

Techniques – Total system cost.

1. INTRODUCTION

The absence of an electrical network in remote regions and the prohibitively high

connection cost due to large distances and irregular topography lead often the various

organizations to explore alternative solutions [1].

Stand alone photovoltaic (SAPV) systems are increasingly viable and cost effective

candidates for providing electricity to remote areas. Such as the ones found in some

remote areas in Algeria. This SAPV system typically consists of a PV array, controller,

battery storage and inverter for AC loads [2].

The successful operation of the SAPV system is to find the optimum relationship

between the PV array and battery storage to meet load demand. Therefore, one optimum

sizing method is essential. The sizing optimization method can help to guarantee the

lowest investment with full use of the renewable energy systems [3].

S. Kebaïli et al.

180

In the literature, various techniques of sizing optimization are used of SAPV system

can be applied to reach a techno-economically optimum. They must search an optimum

combination of two factors: the system cost and power reliability. The relation-ship

between the system cost and reliability should be closed studied, so that an optimum

solution can be reached.

In our previous works [4], we presented the current status of research on optimum

sizing of SAPV systems in period of (1981-2013) and we found a variety of methods

such as intuitive, numerical and analytical. The numerical methods have the advantage

of being more precise and accurate than the others methods. The drawbacks of these

methods are the long calculation times required and the need for long term time series

data of solar radiation.

In short, the numerical methods are the more accurate but also the more difficult to

put into practice. Analytical methods which use equations to describe the PV system

size as a function of reliability. The advantage of these methods is that they combine

accuracy and simplicity. They allow the designer to optimize the energy and economic

cost of the PV system [5].

The merit of a SAPV system should be judged in terms of the reliability of the

electricity supply to the load. This is usually quantified by the concept of loss of load

probability (LOLP) [6]. This concept is defined as the relation-ship between the energy

deficit and the energy demand, as referred to the load, during the total operation time of

the installation [7].

Many of the analytical methods employ this type of methodology for sizing SAPV

systems. In [6], a variety of numerical and analytic models for calculating the LOLP are

described and evaluated using data for three different climatic condition in Spanish

locations (Madrid, Murcia and Santander). For each location, the analytic model

requires as input four different coefficients.

The author in [8] presents a model for the LOLP derived by approximating the

probability density function of the difference between the daily PV array outputs and

the load with two events and by assuming the daily storage charge/discharge process

can be represented as a one step Markov process.

In [9], a simple analytical method which allows one to predict the fraction of the

load covered by a photovoltaic system as a function PV array area, battery storage

capacity, meteorological parameters and the user’s load. These methods are based on

the graphical information of iso reliability lines. The shape of these lines makes it

possible to describe the optimal sizing of system components in an analytic form [10].

In addition to the sizing methods, it’s essential for the designers to choose an

appropriate optimization technique and take into account the most influential parameters

which are suitable for the system sizing. Many optimal sizing techniques were

developed based on the worst month scenario [6, 11]. Yearly average month method

and worst month method were investigated in [11]. Typical meteorological year data or

long period meteorological data were employed by [12].

In this paper, we propose an analytical method to sizing SAPV systems. Firstly, the

LOLP is calculated for different size combinations of PV generator area and useful

battery storage capacity. Secondary, for the desired LOLP at the given daily energy

load, the optimal size combination is obtained at the minimum total system cost at eight

selected sites located in Algeria (Algiers, Oran, Chlef, Tlemcen, Laghouat, Ain Sefra,

Tamanrasset and Tindouf). Finally, the impact of different parameters on the system

size is analysed. The data used in this study were collected from various meteorological

locations in Algeria.

Optimal sizing of a stand-alone photovoltaic systems under various weather…

181

2. PROPOSED ANALYSED METHOD

Due to unavailability of long term hourly data, the monthly mean daily solar

radiation data is used based on worst month scenario. This method is based on the ideas

proposed by Barra et al. [13].

The size of SAPV systems is a general concept including the dimensions of the

photovoltaic generator capacity ‘ AC ’ and the battery storage capacity ‘ SC ’. The

photovoltaic generator capacity, AC , is defined as the ratio of the average daily energy

produced by the generator to the average daily energy demand.

The battery storage capacity, SC , is the ratio of the maximum energy that can be

extracted from the battery or the useful battery storage capacity ‘ UC ’ to the average

daily energy demand. The battery storage capacity represents the number of days of

autonomy in which battery can supply the required energy to load without receiving any

energy from the photovoltaic generator [10].

The expressions of AC and SC are given by [14]:

L

G.A.C m

A

(1)

LCC US (2)

where, is defined follows [13]:

ondbatMPTTPV ... (3)

UC is expressed as found in [7]:

maxbatU DOD.CC (4)

where, L , average daily energy demand (Wh/day); A , area of the photovoltaic

generator (m2); mG , monthly average daily incident solar radiation on the generator

surface, (Wh/m2.day); batC , nominal capacity of the battery (Wh); UC , useful battery

storage capacity (Wh/day); maxDOD , maximum depth of discharge of battery. PV ,

MPPT , bat and ond are the efficiencies of PV generator, the power tracker, the

battery storage and the inverter respectively.

The explicit analytical formulas which relates the average fraction of the energy load

covered by the SAPV system during the mth month, my , with their size is expressed by

[13]:

)y1()yC( mmA (5)

This equation represents a hyperbola curve whose asymptotes are the straight lines:

Am Cy and 1ym (6)

The first means that, for small size SAPV system, all the energy produced by the PV

generator is transferred to the load. The second comes from the consideration that, for

large field areas, the energy supplied by the generator is always able to satisfy the load

[6].

S. Kebaïli et al.

182

Where is free parameter depending on the battery storage capacity. Their

proposed model is given by [13]:

ba (7)

batmtmS ..K.C (8)

Where, m , monthly average of the fractional length of the day from sunrise time to

sunset time. tmK , monthly average clearness index. a and b , constant parameters.

2.1 LOLP curves

All sizing method requires certain level of reliability that the consumer will tolerate

the breakdown of power supply. In this study, the reliability of the system is expressed

in terms of loss of load probability (LOLP) index by means of LOLP curves.

The LOLP curve represents different pairs of CA and CS values which lead to the

same value of LOLP for many Algeria locations. The most universal values of LOLP

range from 12 10LOLP10 in domestic application to 410LOLP in

telecommunication applications [5, 7].

The LOLP curves are expressed by:

)y1(

)C...K(.ayC

m

bsbatmtm

mA

(9)

Once the LOLP curves are obtained, it’s very simple to design both the capacity of

the generator and battery storage capacity.

2.2 System cost

For the desired LOLP, there are many pair’s sets ( sC , AC ), the optimal size

combination is obtained at the minimum system cost.

The cost function of the SAPV system is defined as [2]:

0sAsys CC.C.C (10)

Where, sysC , total cost of the SAPV system; 0C , total constant cost including the

costs of the controller with MPPT, inverter and installations (US$); , unit cost of the

photovoltaic field (US$/Wp); , unit cost of battery storage (US$/Wh).

Most of the existing methods found in the literature [5, 6, 15] determined the values

of AC and SC . The formulated analytical equations for AC and SC of the proposed

method can also be expressed in terms of PV generator area ( A ) and useful battery

storage capacity ( UC ) from {Eq. (1)} and {Eq. (2)} respectively. Furthermore, the cost

function can be expressed in terms of A and UC :

0Um

sys CL

C.

L

G.A..C

(11)

The minimum cost can be found by equating to zero the derivative of the total cost.

So, the condition to obtain the optimum solution of (11) is:


183

mU G..Cd

Ad

(12)

{Eq. (12)} is integrated and the analytical relationship between A and UC is

written as:

kCG..

A Um

(13)

The solution of (12) can be solved graphically in the way that the two curves will be

drawn in the UCA coordinate system. One curve represents different size

combinations of the ( UC , A ) for the given LOLP. The other curve is the straight lines

with a slope of

mG.. which are defined by {Eq. (13)}. The tangent point of

the two curves corresponds to the optimum size of A and UC as shown in figure 1.

Fig. 1: LOLP curve and cost line with various combinations of A and UC

Mathematically, the minimum size ( opt,UC , optA ) is an intersection of a the cost

line with the desired LOLP curve for a fixed value of k . Therefore, the value of k is

obtained as follows:

Um

bU

mm

bm

m

m CG..

C)L/G.(.)y1(

J.a

)L/G.(

yk

(14)

With

L

..KJ batmtmm

(15)

The optimal value of opt,UU CC is calculated by equating to zero the derivative

of k . So, the expression calculated of opt,UC is:

)b11(

mm

mb

mopt,U

)L/G.(..)y1(

G...J.b.aC

(16)

S. Kebaïli et al.

184

The optimal value of optkk is obtained by putting the value of opt,UC from

{Eq.(16)} in {Eq.(14)}.

Finally, substituting both the values of optk and opt,UC into {Eq.(13)}, the

optimum PV generator area ( optAA ) is expressed as:

)b11(

mm

mb

m

mm

bm

m

mopt

)L/G.(..)y1(

G...J.b.a

)L/G.(.)y1(

J.a

)L/G.(

yA

(17)

After getting the optimal point ( opt,UC , optA ), the optimum number of PV module

can be calculated by dividing the optA by the area of a single PV module. Similarly,

the optimum number of batteries can be determined by dividing the opt,UC by the

capacity of a single battery [10].

3. CASE STUDY BASED ON

ALGERIA WEATHER CONDITIONS

The optimization of a SAPV system based on various climatic zones in Algeria is

given to show the application of the above introduced method.

The parameter used to define the different climatic zones is the monthly average

clearness index for all the considered sites. There exist four zones for different climatic

conditions in Algeria which are bound in the following limits [16]:

Zone I 548.0Ktm

Zone II 609.0K548.0 tm

Zone III 671.0K609.0 tm

Zone IV 671.0Ktm

In this study, the database of monthly average daily solar radiation on the

photovoltaic field of the worst month and the monthly average clearness index are

available corresponding to the following sites in [16, 17].

The geographical data for the eights sites located in Algeria is shown in Table 1. As

it can be observed in this table, the locations chosen have different climates and are

widespread over diverse latitudes. The climatic zones repartition for the selected sites

are:

Zone I Algiers, Oran

Zone II Chlef, Tlemcen

Zone III Laghouat, Ain Sefra

Zone IV Tamanrasset, Tindouf.

The daily energy demand is assumed to be constant over the year. The mean daily

energy demand of 1000 Wh/day for domestic purpose was considered.

3.1 Impact of power reliability on system configuration

Applying the previous explained methodology many different LOLP curves have

been calculated.


185

Table 1: Geographical data for the selected sites [17]

Name of site Latitude (°) Longitude (°) Altitude (m)

Algiers 36.43 N 3.15 E 25

Oran 35.38 N 0.70 W 99

Tamanrasset 22.47 N 5.31 E 1378

Tlemcen 34.52 N 1.19 W 810

Ain Sefra 32.45 N 0.34 W 1072

Tindouf 27.40 N 8.09 W 402

Laghouat 33.38 N 2.52 E 767

Chlef 36.08 N 1.17 E 112

The simulation results, related to the system configurations that guarantee the

desired LOLP’s (0.01, 0.05 and 0.1) for some of the locations studied (Algiers, Chlef,

Laghouat, Tamanrasset) are shown in figure 2.

Fig. 2: LOLP curves for Algiers, Chlef, Laghouat and Tamanrasset

For each location, the curves are hyperbolic nature. Each point of them represents

the system size ( SC , AC ) that guarantee the desired LOLP value. Different values of

LOLP are used varying from 0.01 to 0.1. It’s obvious that the PV generator capacity

size significantly decrease when the given LOLP value is taken higher.

This diminution depends on the site. Based on these results, the curves can be

divided in two intervals and their limits depend on the site. In the first interval, by

increasing a battery storage capacity, there is remarkable PV generator capacity

decrease, while in the second interval, the PV generator capacity decreases gradually

and remains almost constant, with the increase of battery storage capacity.

S. Kebaïli et al.

186

Figure 3 shows the LOLP curves for the locations considered in this study, with

LOLP value of 0.01. There are differences in the PV generator capacity for the different

days of autonomy considered depending on whether the locations are Algiers or

Laghouat in Fig. 3a and Oran or Ain Sefra in Fig. 3b. This confirms that the locations

belong to different climatic areas. According to these results, the size of PV generator

capacity almost more important at the sites of Algiers and Oran than the other sites,

while at Laghouat and Ain Sefra sites is relatively low.

-3a- -3b-

Fig. 3: LOLP curves for differents locations for LOLP=0.01

This can be explained by that the locations of Algiers and Oran or Laghouat and Ain

Sefra belong to same climatic areas.

The results obtained for the previous mentioned locations with a LOLP value of 0.05

are illustrated in Figure 4. In this case, the PV generator capacity are more decrease

according to location than in the case of LOLP=0.01 which is confirm by {Eq. (9)}.

-4a- -4b-

Fig. 4: LOLP curves for differents locations for LOLP=0.05

For all the considered sites, the different values of LOLP for different size

combinations of useful battery storage and PV generator area is illustrated in figure 5.

The simulation results show that the system sizing ( UC , A ) depends on the system

reliability.


187

We can be seen that the LOLP is inversely proportionel to the system sizing. With

high LOLP values, the PV generator area increases with the LOLP diminution. For

lower LOLP values, a considerable increase in the PV generator area in encountered

with LOLP diminution in all the considered sites.

3.2 Impact of function cost on system configuration

The function cost is utilized as an economic criteria, the optimal system

configuration is the one which has the lowest cost.

At constant load and given LOLP (LOLP = 0.01), the optimal PV generator area

( optA ), optimal useful battery storage ( opt,UC ) and optimal total cost of SAPV system

without taking consideration total constant costs ( 0opt,sysopt CCC ) were

calculated at different values of and . The simulation results are presented in Table

2. According to these results, we can be seen that:

If is constant and varied: increase of decrease the value of UC and vice

versa ( = 8 and = 0.2, 0.6).

If is constant and varied: decrease of α increases the value of A and vice versa

( = 0.2 and = 5, 8).

If the optimum cost at and is calculated, ),(C 11opt , the optimum cost at

1 = x. and = x. is ),(C 11opt = x. ),(Copt Case: = 4 and =

0.3, = 8 and = 0.6.

In general, the southern locations show lower optimum size ( opt,UC , optA ), as

Tindouf and Tamanrasset, than the locations of the north, as Algiers and Oran,

whose solar potential is lower.

-5a- -5b-

Fig. 5: Different size combination of useful battery storage and

PV generator area at different LOLP values

Figure 6 shows the impact of the LOLPs on the optimal cost of energy for the

considered sites. It’s clear that the optC is found to be sensitive to the desired LOLPs.

It’s observed that with a value of 0.05 instead of 0.01, the optimal cost of energy

will get reduced for the different sites. Ain Sefra, Laghouat and Tindouf have the lowest

optimal cost than the others sites.

S. Kebaïli et al.

188

Table 2: Optimal system configuration at different values of and

Nom Site opt,UC optA optC

Algiers 4 0.3 11931 3.1076 0.91024

5 0.2 14443 2.7959 09.1000

8 0.6 11931 3.1076 18.2047

8 0.2 16661 2.6375 12.7073

Oran 4 0.3 11604 3.1111 08.9613

5 0.2 14047 2.8052 08.9862

8 0.6 11604 3.1111 17.9226

8 0.2 16204 2.6498 12.5760

Chlef 4 0.3 10749 2.8392 08.5929

5 0.2 13012 2.5753 0.8.6888

8 0.6 10749 2.8392 17.1850

8 0.2 15010 2.4411 12.2330

Tlemcen 4 0.3 10426 3.2098 08.4536

5 0.2 12621 2.9181 08.5764

8 0.6 10426 3.2098 16.9071

8 0.2 14559 2.7698 12.1033

Laghouat 4 0.3 09349.2 2.3277 07.9895

5 0.2 11318.0 2.1328 08.2020

8 0.6 9349.2 2.3277 15.9791

8 0.2 13056.0 2.0337 11.6713

Ain Sefra 4 0.3 09026 2.2088 07.8502

5 0.2 10926 2.0288 08.0895

8 0.6 09026 2.2088 15.7005

8 0.2 12604 1.9373 11.5416

Taman. 4 0.3 09907.7 2.1598 08.2303

5 0.2 11994.0 1.9708 08.3962

8 0.6 09907.7 2.1594 16.4605

8 0.2 13836.0 1.8748 11.8954

Tindouf 4 0.3 09439.7 2.5861 08.0286

5 0.2 11427.0 2.3680 08.2334

8 0.6 09439.7 2.5861 16.0571

8 0.2 13182.0 2.2571 11.7077

Fig. 6: Impact of LOLP on the optimal cost of energy


189

3.3 Impact of load on system configuration

At various daily energy load (500, 1000 and 1500 Wh/day) with and constant,

the PV generator area and useful battery storage capacity is illustrated in figure 7. It’s

obvious that the increase of load will increase the PV generator area and useful battery

storage capacity, corresponding to the optimal combination.

Figure 8 presented the optimum cost ( optC ), as a function of daily energy load for

different sites. It’s clear that, the optimum cost is remaining constant with increase in

the load. This figure shows that Algiers site has the higher cost than the others sites.

Contrary, Ain Sefra site has the lower cost.

Fig. 7: Impact of load on the LOLP curves for considered sites

S. Kebaïli et al.

190

Fig. 8: Impact of load on the optimum cost for considered sites

4. CONCLUSION

This paper presents the sizing and techno-economical optimization of a SAPV

systems for eight sites located in Algeria. A proposed analytical method has been

developed for the optimal sizing based on reliability and cost by using monthly average

daily data of solar radiation in the worst weather conditions.

According to the results, related to the considered sites, it can be concluded that:

The optimal configuration system size depends on the site.

The computed optimal PV generator area and the useful battery storage capacity

were calculated at the tangent point of the desired LOLP curves. In general, the

southern locations show lower optimum size ( optUC , optA ) than the locations of

the north (case of Tindouf and Tamanrasset)

For all considered sites, the optimum combination increase with increase in load.

At Ain Sefra site, the optimal cost ( optC ) is found to be the lowest due to the high

available solar potential. Hence, the system size is reduced.

For all considered sites, the optimal cost is remains constant by change in load.

This proposed analytical method can be applied to any locations taking account of

the weather conditions, clearness index, efficiencies and costs of the components of size

optimization of SAPV system.

REFERENCES

[1] S. Diaf, G. Notton, M. Belhamel, M. Haddadi and A. Louche, ‘Design and Techno-

Economical Optimization for Hybrid PV/Wind System under Various Meteorological

Conditions’, Applied Energy, Vol. 85, N°10, pp. 968 - 987, 2008.

[2] W.X. Shen, ‘Optimally Sizing of Solar Array and Battery in a Stand Alone Photovoltaic

System in Malaysia’, Renewable Energy, Vol. 34, N°1, pp. 348 - 352, 2009.

[3] A. Kaabeche, M. Belhamel and R. Ibtiouen, ‘Techno-Economic Valuation and Optimization of

Integrated Photovoltaic/Wind Energy Conversion System’, Solar Energy, Vol. 85, N°10, pp.

2407 - 2420, 2011.

[4] S. Kebaili and H. Benalla, ‘Optimal Sizing of Stand Alone Photovoltaic Systems: A Review’,

Accepted in Journal of Electrical Engineering, 2014.


191

[5] L. Hontoria, J. Aguilera and P. Zufiria, ‘A New Approach for Sizing Stand Alone Photovoltaic

Systems Based in Neural Networks’, Solar Energy, Vol. 78, N°2, pp. 313 - 319, 2005.

[6] M. Egido and E. Lorenzo, ‘The Sizing of Stand Alone PV Systems: A Review and a Proposed

New Method’, Solar Energy Materials and Solar Cells, Vol. 26, N°1-2, pp. 51 - 69, 1992.

[7] R. Posadillo and R. Lopez Luque, ‘Approaches for Developing a Sizing Method for Stand

Alone PV Systems with Variable Demand’, Renewable Energy, Vol. 33, N°5, pp. 1037 -

1048, 2008.

[8] L. Bucciarelli, ‘Estimating Loss of Power Probabilities of Stand Alone Photovoltaic Solar

Energy Systems’, Solar Energy, Vol. 32, N°2, pp. 205 - 209, 1984.

[9] B. Bartoli, V. Cuomo, F. Fantana, C. Serio and V. Silverstrini, ‘The Design of Photovoltaic

Plants: An Optimization Procedure’, Applied Energy, Vol. 18, N°1, pp. 37 - 41, 1984.

[10] A.Q. Jakhrani, A.K. Othman and A.R.H. Rigit and S.R. Samo, ‘A Novel Analytical Model for

Optimal Sizing of Stand Alone Photovoltaic Systems’, Energy, Vol. 46, N°1, pp. 675 - 682,

2012.

[11] C. Protogerolous, B.J. Brinkworth and R. Marshall, ‘Sizing and Techno-Economical

Optimization For Hybrid Solar PV-Wind Power System with Battery Storage’, International

Journal of Energy Research, Vol. 21, N°6, pp. 465 - 479, 1997.

[12] E. Koutroulis, ‘Methodology for Optimal Sizing of Stand-Alone Photovoltaic/Wind

Generator Systems using Genetic Algorithms’, Solar Energy, Vol. 80, N°9, pp. 1072 – 1088,

2006.

[13] L. Barra, S. Catalanotti, F. Fontana and Lavorante, ‘An Analytical Method to Determine the

Optimal Size of a Photovoltaic Plant’, Solar Energy, Vol. 33, N°6, pp. 509 - 514, 1984.

[14] R. Posadillo and R. Lopez Luque, ‘A Sizing Method for Stand Alone PV Installations with

Variable Demand’, Renewable Energy, Vol. 33, N°3, pp. 1049 - 1055, 2008.

[15] M. Sidrach-de-Cardona and M.L. Li, ‘A Simple Model for Sizing Stand Alone Photovoltaic

Systems’, Solar Energy Materials and Solar Cells, Vol. 55, N°3, pp. 199 - 214,1998.

[16] A. Hadj Arab, B. Ait Driss, R. Amimeur and E. Lorenzo, ‘Photovoltaic Systems Sizing for

Algeria’, Solar Energy, Vol. 54, N°2, pp. 99 - 104, 1995.

[17] A. Mellit, S.A. Kalogirou, S. Shaari and A. Hadj Arab, ‘Methodology for Predicting

Sequences of Mean Monthly Clearness Index and Daily Solar Radiation Data In Remote

Areas: Application For Sizing a Stand Alone PV System’, Renewable Energy, Vol. 33, N°7,

pp. 1570 - 1590, 2008.

optimal sizing of a stand-alone photovoltaic systems

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